Coffee Cake with Coffee Icing

3/4 cup milk
2 heaping teaspoons instant coffee
1 teaspoon vanilla
3 cups cake flour
2 1/2 teaspoons baking powder
1/2 teaspoon baking soda
1 teaspoon salt
2 teaspoons very finely ground coffee
2 teaspoons unsweetened natural cocoa powder
3/4 cup butter at room temperature, cut into cubes
1 1/2 cup white sugar
4 eggs, yolks and whites separated

Coffee Glaze
1/4 cup butter, at room temperature
2 cups powdered sugar
3 tablespoons freshly brewed coffee

Chocolate curls, for decoration